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In the attached page http://www.geocities.com/complementarytheory/no1.pdf we can understand that there is a deep connection between our abstract ideas and the ways that they are represented by us. I'll be glad to know your point of view.

Cantor's second diagonal method is not from N to R because each element in the list is only a non-accurate representation of R member. Please open: http://www.geocities.com/complementarytheory/NEXT.pdf Thank you.

Let us say that your own life depends on your ability to explain the Bijection concept to a 5 years old child. Please write your explanation. Thank you.

Cantor's second diagonal method is not form N to P(N) because each element in the list is only a non-accurate representation of R member, for example: Epsilon = Invariant Proportion About 3.14... = circumference/diameter: Let us say that Epsilon is equivalent to the invariant proportion that can be found in the triangles below. (VERY IMPORTANT: When Epsilon = Invariant Proportion, then there is no connection to words like 'smaller' or 'bigger' or 'size' or 'magnitude' or 'Quantity', and the reason is clearly explained) , |\ | \ | \ | \ | | | |\ | | \ | | \ | | \ | | | | | |\ | | | \ | | | \ | | | | | | | |\ | | | | \ | | | | | | | | | |\ |____|____|___|__|_\ Each arbitrary right triangle's area is smaller than any arbitrary left triangle's area, but the internal proportion of each triangle remains unchanged, so it does not depend on size or magnitude (please think about circumference/diameter ratio, which does not depend on a circle's size). If we have finitely many triangles then this proportion can be found finitely many times. But in the case of infinitely many triangles, this proportion can be found infinitely many times. Since Epsilon is equivalent to this proportion, it cannot be found if and only if this proportion cannot be found. It is clear that if the proportion can be found infinitely many times, than it cannot be eliminated, and if it is eliminated, it means that it is found only finitely many times. In other words, any collection of infinitely many elements can be found if and only if some epsilon that belongs to it also can be found, and if this Epsilon cannot be found, then there are only two options, which are: a) The collection does not exist. b) The collection is a finite collection. Conclusion: There is an inseparable connection between the PERMANENT EXISTENCE of an epsilon and the collection of infinitely many elements that is related to it. In other words, there is no way to calculate the exact SUM of infinitely many elements, because the SUM of infinitely many elements cannot be more than SUM – epsilon, and therefore the accurate SUM of infinitely many elements does not exist. Therefore 3.14... < The accurate value of circumference/diameter. So we can clearly see that any Base # representation of R member is always < R member. Conclusion: Cantor’s second diagonal method is not between N and P(N) but between N and N. In this case we can immediately use the diagonal number in order to proof that the cardinality of a non-finite collection cannot be found because: The map f(z)=z-(diagonal number) is not a bijection from Z to Z.

Matt please read carefully post #64

Ok, If Matt cannot do it, then let us continue. Now Matt as I clearly explained, the existence of a non-finite collection depends on the 'existence of the next element' as we can clearly show, by using Cantor's second-diagonal method. In a non-finite collection, the diagonal number must be permanently added to the collection. In a finite collection, the diagonal number must not be added to the collection. We clearly know that we cannot define a bijection in the second diagonal method, because each time we define a bijection, we discover that there is a new element which is out of the domain of our 1-1 correspondence mapping (a permanent next element). Conclusion: No bijection can be found in a non-finite collection, and this property can be found in any given non-finite collection and only in a non-finite collection. Therefore the cardinality of a non-finite collection, like set N (for example) is |N|-NEXT.

A very simple question that needs a very simple answer.

Bloodhond are you also Matt Grime? I am still waiting to Matt's very simple English exaplanation of the Bijection concept.

No Blarg, I am waiting to Matt's explanation of the bijection concept, in a very simple English.

Matt, Please explain in a very simple English what is a bijection?

Any given collection is identical to itself. The existence of a non-finite collection depends on the 'existence of the next element' as we can clearly show, by using Cantor's second-diagonal method. In a non-finite collection, the diagonal number must be permanently added to the collection. In a finite collection, the diagonal number must not be added to the collection.

Dear Matt, The existence of the next has nothing to do with order. The only notion by these notations is: Any given collection is identical to itself.

Dear Matt, Finite collection and non-finite collection cannot be joining together to a one class, as Cantor tried to do. And the reason is very simple. In a finite collection the successor>0 is not a permanent property, and because of this reason we can define the Cardinal of a finite collection. But in a non-finite collection a successor>0 is a permanent property, and because of this reason we cannot define the Cardinal of a non-finite collection. Actually we can use Cantor’s diagonal method in order to prove it. The diagonal number which is not in the list, simply proves that we cannot define the complete list of a non-finite collection, because this diagonal number is actually the permanent next element that cannot allowed us to define a complete collection of infinitely many elements. Cantor did not understand its own diagonal method, because he used the hidden assumption that such a complete list of non-finite collection can exist, by ignoring the fundamental difference that exists between a finite collection and a non-finite collection. And this fundamental difference is reduced to one and only one property, which is: The existence of the next. The permanent existence of the next is a fundamental property of a non-finite collection. The non-permanent existence of the next is a fundamental property of a finite collection. Very important: We cannot define a one class for both of them because there is a XOR connectivity between a finite collection and a non-finite collection that can be written as: Finite collection XOR Non-finite collection. Yours, Doron